Do you must open a mail client app to get notifications?

I noticed today you must have the mail app open to get notifications for new emails. This behavior seems to contradict iPhone's because any email app on the iPhone runs in the background so without you having to open the app, you get a notification for new emails automatically. But on Mac OS that doesn't seem like the case. No email client app on Mac OS, whether it's Apple's mail app or 3rd party app, do not run in the background so you can't get notifications or whatsoever unless you manually open the app first.

I just don't want to have a mail app on my dock because it takes up space. Am I missing something or is there an easy fix for this? Any help would be much appreciated!!